DE 10 2007 013 698 A1 and the post-published DE 10 2009 022 545 A1 disclose thermoformer packaging machines as a special type of packaging machines. In the case of these thermoformer packaging machines, packaging troughs are thermoformed in a bottom foil, filled with a product and subsequently sealed with a cover film. After having been sealed, the packages must be singulated. To this end, cutting units are normally provided in the longitudinal direction and in the transverse direction of the working direction of the packaging machine. These cutting units are subjected to high loads, since they execute fast movements and since they must exert strong forces for cutting through packaging foils, in particular if these packaging foils are thick. Hence, they are liable to fatigue and regular maintenance is necessary. In addition, conventional cutting units often consume a lot of energy and take up much space.